First we note that under
\(\widehat{\mathop {\rm U{}}\nolimits }(\mathcal {H})\cong \mathop {\rm U{}}\nolimits (\mathcal {H})\)  the spin representation decomposes
into the irreducible subspaces \(\Lambda ^n(\mathcal {H})\)  with
\( {\pi _a}(z,g)(v_1 \wedge \cdots \wedge v_n)= z gv_1 \wedge \cdots \wedge g v_n \quad \mbox{ for } \quad v_1,\ldots , v_n \in \mathcal {H}. \) 
If \(i x \le 0\) , then the corresponding operator
on \(\Lambda ^n(\mathcal {H}) \subseteq \mathcal {H}^{\otimes n}\)  is also
\(\le 0\) , and this implies that
\( s_{\pi _a}(t,x) = -t + s_{\pi _a}(x) \le - t < \infty \quad \mbox{ for } \quad x \in C_{{\mathfrak {u}}(\mathcal {H})}. \) 
Writing \({\mathfrak {z}}\cong {\mathbb {R}}\)  for the center of \(\widehat{{\mathfrak {o}}}_{\rm res}(\mathcal {H})\) ,
we thus find that
\({\mathfrak {z}}\times C_{{\mathfrak {u}}(\mathcal {H})} \subseteq B(I_{\pi _a}).\)
Next we consider the decomposition of \({\mathfrak {g}}= \widehat{{\mathfrak {o}}}_{\rm res}(\mathcal {H})= {\mathfrak {k}}\oplus {\mathfrak {p}}_2\) , where \({\mathfrak {k}}= {\mathfrak {z}}\times {\mathfrak {u}}(\mathcal {H})\)  and
\({\mathfrak {p}}_2 = \lbrace  x \in {\mathfrak {o}}_{\rm res}(\mathcal {H}) \colon  Ix = - xI\rbrace .\) 
For the map
\( F \colon  {\mathfrak {k}}\times {\mathfrak {p}}_2 \rightarrow {\mathfrak {g}}, \quad (x,y) \mapsto e^{\mathop {{\rm ad}}\nolimits y} x \quad \mbox{ we then have }\quad {\tt d}F(x,0)(a,b) = a + [b,x], \) 
which is invertible if and only if \(\mathop {{\rm ad}}\nolimits x \colon  {\mathfrak {p}}_2 \rightarrow {\mathfrak {p}}_2\)  is
invertible. This is in particular the case for
\(x = I\) , the complex structure of \(\mathcal {H}\) . Therefore the image of
\(({\mathfrak {z}}\times C_{{\mathfrak {u}}(\mathcal {H})}) \times {\mathfrak {p}}_2\)  has interior points,
and since it is contained in the invariant subset
\(B(I_{\pi _a})\) , the cone
\(W_{\pi _a} = B(I_{\pi _a})^0\)  is non-empty.
With Proposition REF  we now see that the
representation \((\pi _a, \Lambda (\mathcal {H}))\)  is semibounded.
To determine the cone \(W_{\pi _a}\) , we recall from Remark REF 
that for each \(x \in {\mathfrak {o}}_1(\mathcal {H})\) , the corresponding
operator \({\tt d}\pi _a(x)\)  on \(\Lambda (\mathcal {H})\)  is bounded. This means that
\({\mathfrak {o}}_1(\mathcal {H}) \subseteq B(I_{\pi _a}) \subseteq \overline{W_{\pi _a}}\) , and
since \(H(W_{\pi _a})\)  is a closed ideal (Lemma REF (v)),
it follows that
\( {\mathfrak {n}}:= {\mathfrak {z}}\oplus {\mathfrak {u}}_\infty (\mathcal {H}) \oplus {\mathfrak {p}}_2 \subseteq H(W_{\pi _a}). \) 
In view of \(\widehat{{\mathfrak {o}}}_{\rm res}(\mathcal {H}) = {\mathfrak {u}}(\mathcal {H}) + {\mathfrak {n}}\) , this proves that
\(W_{\widehat{{\mathfrak {o}}}_{\rm res}(\mathcal {H})} = C_{{\mathfrak {u}}(\mathcal {H})} + {\mathfrak {n}}\subseteq W_{\pi _a}= {\mathfrak {n}}+ (W_{\pi _a} \cap {\mathfrak {u}}(\mathcal {H})). \) 
It therefore remains to show that
\( W_{\pi _a} \cap {\mathfrak {u}}(\mathcal {H}) \subseteq C_{{\mathfrak {u}}(\mathcal {H})} + {\mathfrak {u}}_\infty (\mathcal {H}). \) 
To verify this assertion, let \(A = A^*\)  be a hermitian
operator on \(\mathcal {H}\)  and \({\tt d}\pi _a(A)\)  denote the corresponding operator
on \(\Lambda ^n(\mathcal {H})\) . Suppose that there exists an \(\varepsilon > 0\) 
such that the range of the spectral projection
\(P([\varepsilon ,\infty [)\)  is infinite dimensional. Then there exists
for each \(n\)  an orthonormal subset \(v_1, \ldots , v_n\)  in this space.
We then have
\(&\ \ \ \ \langle {\tt d}\pi _a(A)(v_1 \wedge \cdots \wedge v_n),v_1 \wedge \cdots \wedge v_n \rangle \\&= \sum _{j = 1}^n \langle v_1 \wedge \cdots \wedge Av_j \wedge \cdots \wedge v_n,v_1 \wedge \cdots \wedge v_n \rangle = \sum _{j = 1}^n \langle Av_j, v_j \rangle \ge n \varepsilon ,\) 

and this implies that \({\tt d}\pi _a(A)\)  is not bounded from above.
If, conversely, for every \(\varepsilon > 0\) 
the spectral projection \(P([\varepsilon ,\infty [)\)  has finite dimensional
range, then \(A = A_- + A_+\) , where
\(A_- \le 0\)  and \(A_+\)  is compact, i.e.,
\(iA \in \overline{C_{{\mathfrak {u}}(\mathcal {H})}} + {\mathfrak {u}}_\infty (\mathcal {H})\) .
This implies that the open cone \(W_{\pi _a} \cap {\mathfrak {u}}(\mathcal {H})\) 
is contained in \(C_{{\mathfrak {u}}(\mathcal {H})} + {\mathfrak {u}}_\infty (\mathcal {H})\) .
\(\Box \)
Remark 10.8  
(a) With a similar argument as in Remark REF  ones
argues that under the action of the identity component
\(\widehat{\mathop {\rm O{}}\nolimits }_{\rm res}(\mathcal {H})_0\)  on the space \(\Lambda (\mathcal {H})\) 
decomposes into two irreducible subrepresentations
\(\Lambda ^{\rm even}(\mathcal {H})\)  and \(\Lambda ^{\rm odd}(\mathcal {H})\) .
However, the action of the full group \(\widehat{\mathop {\rm O{}}\nolimits }_{\rm res}(\mathcal {H})\) 
is irreducible because the elements \(g \in \widehat{\mathop {\rm O{}}\nolimits }_{\rm res}(\mathcal {H})\) 
not contained in the identity component exchange
the two subspaces
\(\Lambda ^{\rm even}(\mathcal {H})\)  and
\(\Lambda ^{\rm odd}(\mathcal {H})\)  ([1]}).
(b) There is also an analog of the Banach–Lie algebra
\(\mathfrak {hsp}_{\rm res}(\mathcal {H})\)  acting irreducibly on \(\Lambda (\mathcal {H})\) 
in the fermionic case. Here it is an infinite dimensional
analog of the odd orthogonal Lie algebra \(\mathop {{\mathfrak {so} }}\nolimits _{2n+1}({\mathbb {R}})\) .
To see this Lie algebra, we define linear and antilinear
rank-one operators on \(\mathcal {H}\)  by
\( P_{v,w}(x) := \langle x,w \rangle v \quad \mbox{ and } \quad \overline{P}_{v,w}(x) := \langle w,x\rangle v \) 
and observe that \(P_{v,w}^* = P_{w,v}\)  and
\(\overline{P}_{v,w}^* = \overline{P}_{w,v}\) .
Therefore \(Q_{v,w} := P_{v,w} - P_{w,v} \in {\mathfrak {u}}_1(\mathcal {H})\) ,
and a direct calculation
yields
\( {\tt d}\pi _a(Q_{v,w}) = a_0^*(v)a_0(w) - a_0^*(w)a_0(v). \) 
As \(\mathop {{\rm tr}}\nolimits (Q_{v,w}) = 2i \mathop {{\rm Im}}\nolimits \langle v,w\rangle \) , we obtain with the map
\(\sigma \)  from (REF ) (Remark REF )
\( {\tt d}\pi _a(\sigma (Q_{v,w})) = {\tt d}\pi _a(-\mathop {{\rm Im}}\nolimits \langle v,w \rangle , Q_{v,w}). \) 
For the antilinear operators \(\overline{Q}_{v,w} :=\overline{P}_{v,w} - \overline{P}_{w,v}\)  we have
\(\overline{Q}_{v,w} \in {\mathfrak {o}}_1(\mathcal {H}) \subseteq {\mathfrak {o}}_{1,2}(\mathcal {H})\)  and
\( {\tt d}\pi _a(\overline{Q}_{v,w})= a_0^*(v)a_0^*(w) - a_0(w) a_0(v) = a_0^*(v)a_0^*(w) + a_0(v) a_0(w). \) 
Next we observe that the operators \(\rho (v) := \frac{1}{\sqrt{2}}(a_0(v) -a_0^*(v))\)  satisfy
\(&\ \ \ \ [\rho (v), \rho (w)]= {\textstyle {\frac{1}{2}}}[a_0(v) - a_0^*(v), a_0(w) - a_0^*(w)] \\&= (a_0(v)a_0(w) + a_0^*(v)a_0^*(w))+ (a_0^*(w) a_0(v) - a_0^*(v)a_0(w)) + i\mathop {{\rm Im}}\nolimits \langle v, w \rangle \\&= {\tt d}\pi _a(\mathop {{\rm Im}}\nolimits \langle v, w \rangle , \overline{Q}_{v,w}-Q_{v,w})={\tt d}\pi _a(\sigma (\overline{Q}_{v,w}- Q_{v,w})).\) 

This calculation implies that we obtain on the direct sum
\(\mathcal {H}\oplus {\mathfrak {o}}_{1,2}(\mathcal {H})\)  a Banach–Lie algebra structure
with the bracket
\( [(v,X), (v^{\prime },X^{\prime })]:= (Xv^{\prime } - X^{\prime }v, [X,X^{\prime }] + \overline{Q}_{v,w}-Q_{v,w}), \) 
and
\( (v,X) \mapsto \rho (v) + {\tt d}\pi _a(\sigma (X)) \) 
defines a representation by unbounded operators on \(\Lambda (\mathcal {H})\) ,
where the subalgebra \(\mathcal {H}\oplus {\mathfrak {o}}_1(\mathcal {H})\)  is represented by bounded
operators. A closer inspection shows that
\(\mathcal {H}\oplus {\mathfrak {o}}_1(\mathcal {H}) \cong {\mathfrak {o}}_1(\mathcal {H}\oplus {\mathbb {R}})\)  is an
infinite dimensional version of \({\mathfrak {o}}_{2n+1}({\mathbb {R}})\) 
(cf. [2]}).

Definition 10.9  Let \({\mathcal {I}}_\beta \subseteq \mathop {{\rm GL}}\nolimits (\mathcal {H}_{\mathbb {R}})\)  be the set of
orthogonal real-linear complex structures on the real
Hilbert space \(\mathcal {H}_{\mathbb {R}}\) . This set parameterizes the
complex Hilbert space structures on \(\mathcal {H}_{\mathbb {R}}\)  compatible
with the given real Hilbert space structure.

Lemma 10.10  The following assertions hold:
(i)
\({\mathcal {I}}_\beta = \mathop {\rm O{}}\nolimits (\mathcal {H}) \cap {\mathfrak {o}}(\mathcal {H})= \lbrace  g \in \mathop {{\rm GL}}\nolimits (\mathcal {H}_{\mathbb {R}}) \colon  g^\top = g^{-1} = -g \rbrace \)  is a submanifold
of \(\mathop {\rm O{}}\nolimits (\mathcal {H})\) .
(ii)
The conjugation action of \(\mathop {\rm O{}}\nolimits (\mathcal {H})\)  on
\({\mathcal {I}}_\beta \)  leads to a diffeomorphism \({\mathcal {I}}_\beta \cong \mathop {\rm O{}}\nolimits (\mathcal {H})/\mathop {\rm U{}}\nolimits (\mathcal {H})\) .
(iii)
\({\mathcal {I}}_\beta ^{\rm res} := \mathop {{\rm Ad}}\nolimits (\mathop {\rm O{}}\nolimits _{\rm res}(\mathcal {H}))I= \lbrace  J \in {\mathcal {I}}_\beta \colon  \Vert I - J \Vert _2 < \infty \rbrace \cong \mathop {\rm O{}}\nolimits _{\rm res}(\mathcal {H})/\mathop {\rm U{}}\nolimits (\mathcal {H})\) .


